Related papers: Fractional integration with singularity on Light-c…

In this first part of our project, we prove a classical Hardy-Littlewood-Sobolev result for a new family of fractional integral operators whose kernel has singularity appeared on the light cone in R^n+1.

We study a family of convolution operators whose symbols and kernels have singularity on the light-cone in $\mathbb{R}^{n+1}$. First, we prove a desired ${\bf L}^p\to {\bf L}^q$ norm inequality which has been left open. Moreover, we obtain…

We study a family of convolution operators whose kernels have a singularity on the unit sphere. As a result, we prove the regarding L^p-L^q Sobolev inequalities.

We study a family of fractional integral operators whose kernels satisfying an non-isotropic dilation have singularity on a coordinate subspace. A characterization is given for these operators bounded from the classical, atom decomposable…

We introduce a 2-parameter Hedberg's method, and use it to prove a Hardy-Littlewood-Sobolev theorem for a new type of fractional integral operators, whose kernel has singularity on the light-cone.

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes…

We study a family of convolution operators. Their regarding Fourier multipliers are defined in terms of distributions having singularity on the light-cone in $\mathbb{R}^{n+1}$. As a result, we give a new approach to the Bochner-Riesz…

In this paper, we study a class of multilinear fractional integral operators which have correlation kernels $\prod_{1\leq i<j \leq k}|x_i-x_j|^{-\alpha_{ij}}$. The necessary and sufficient conditions are obtained under which these oprators…

We study a family of strong fractional integral operators whose kernels have singularity on every coordinate subspace. We prove a two-weight $L^p$-$L^q$-norm inequality by allowing only one of the weights to satisfy $A_p\times…

In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…

We study a family of fractional integral operators defined in $\mathbb{R}^3$ whose kernels are distributions associated with Zygmund dilations: $(x_1, x_2, x_3) \rightarrow (\delta_1 x_1, \delta_2 x_2, \delta_1\delta_2 x_3)$ for…

This paper studies fractional integral operator for vector fields in weighted $L^1$. Using the estimates on fractional integral operator and Stein-Weiss inequalities, we can give a new proof for a class of Caffarelli-Kohn-Nirenberg…

This article introduces two new fractional operators with sine ($\sin$) and cosine ($\cos$) kernels, motivated by their fundamental role in modeling AC signals in electrical circuits. The operators are designed to improve the analysis of…

We introduce and study the properties of a new family of fractional differential and integral operators which are based directly on an iteration process and therefore satisfy a semigroup property. We also solve some ODEs in this new model…

In the paper we consider self-adjoint partial integral operators of Fredholm type $T$ with a degenerate kernel on the space $L_2([a,b]\times[c,d]).$ Essential and discrete spectra of $T$ are described.

The paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of general derivative operator acting on the functions analytic on a curve in complex plane is deduced and the…

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented and some…

We study the Hp-Lq boundedness of certain integral operators of fractional type.

In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these…

Inverse spectral problems are studied for first-order integro-differential operators on a finite interval. These problems consist in recovering some components of the kernel from one or multiple spectra. Uniqueness theorems are proved for…